Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2014, Article ID 864989, 6 pages
http://dx.doi.org/10.1155/2014/864989
Research Article
Control System Design of Shunt Active Power Filter Based on
Active Disturbance Rejection and Repetitive Control Techniques
Le Ge,1,2 Xiaodong Yuan,3 and Zhong Yang1
1
College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
School of Electric Power Engineering, Nanjing Institute of Technology, Nanjing 211167, China
3
Department of Renewable Energy and Distribution Network Technology, Jiangsu Electric Power Company Research Institute,
Nanjing 211103, China
2
Correspondence should be addressed to Le Ge; gele nit@sina.com
Received 13 December 2013; Accepted 18 February 2014; Published 23 March 2014
Academic Editor: Xiaojie Su
Copyright © 2014 Le Ge et al. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
To rely on joint active disturbance rejection control (ADRC) and repetitive control (RC), in this paper, a compound control law for
active power filter (APF) current control system is proposed. According to the theory of ADRC, the uncertainties in the model and
from the circumstance outside are considered as the unknown disturbance to the system. The extended state observer can evaluate
the unknown disturbance. Next, RC is introduced into current loop to improve the steady characteristics. The ADRC is used to get
a good dynamic performance, and RC is used to get a good static performance. A good simulation result is got through choosing
and changing the parameters, and the feasibility, adaptability, and robustness of the control are testified by this result.
1. Introduction
The proliferation of nonlinear loads caused by more and more
modem electronic equipments results in deterioration of
power quality in power transmission or distribution systems.
Harmonic, reactive, negative sequence and flickers are the
reasons of various undesirable phenomena in the operation of
power system. In order to solve these problems, the concept
of active power filter (APF) was presented. Active power
filters, which compensate harmonic and reactive current
component for the power supplies, can improve the power
qualities and enhance the reliabilities and stabilities on power
utility [1–3]. In recent 30 years from APF presented, the
continual innovation of control strategies mainly impels the
APF techniques to be developed rapidly [4–7].
Active disturbance rejection control (ADRC) is a robust
control method that is based on extension of the system
model with an additional and fictitious state variable, representing everything that the user does not include in the
mathematical description of the plant [8–11]. Different from
other disturbances and states estimation [12–15], this virtual
state (sum of internal and external disturbances, usually
denoted as a “total disturbance”) is estimated online with
a state observer and used in the control signal in order to
decouple the system from the actual perturbation acting on
the plant. This disturbance rejection feature allows user to
treat the considered system with a simpler model, since the
negative effects of modeling uncertainty are compensated in
real time. As a result, the operator does not need a precise
analytical description of the system, as one can assume the
unknown parts of dynamics as the internal disturbance in
the plant. Robustness and the adaptive ability of this method
make it an interesting solution in scenarios where the full
knowledge of the system is not available.
Repetitive control is a control method developed by a
group of Japanese scholars in 1980s. It is based on the Internal
Model Principle and used specifically in dealing with periodic
signals, for example, tracking periodic reference or rejecting
periodic disturbances. The repetitive control system has been
proven to be a very effective and practical method dealing
with periodic signals [15–18]. Repetitive control has some
similarities with iterative learning control.
This paper addresses the electric current tracking control
problem for shunt APF. The control law is joint ADRC and
RC which can deal with the static and dynamic performance.
The rest of this paper is organized as follows. In Section 2,
2
Mathematical Problems in Engineering
y∗ (t)
Transient
profile
generator
y∗(n) (t)
···
y∗̇ (t)
Nonlinear u0 (t)
weighted
sum
−
1/b(t)
d(t)
u(t)
Plant
y(t)
b(t)
.
.
.
zn
zn+1 = â (t)
Extended state
···
observer (ESO)
z1
Figure 1: Block diagram of the ADRC.
a brief description of the ADRC is presented. In Section 3,
main results of ADRC + RC control technique are developed.
In Section 4, simulation results are presented to show the
effectiveness of the proposed control technique. Finally, some
conclusions are made in Section 5.
2. Active Disturbance Rejection Control
In ADRC, the tracking differentiator (TD) is used to deal with
the reference input and the extended state observer (ESO) is
used to deal with the output of controlled system. Then the
ADRC control law can be selected through the appropriate
nonlinear combination of state errors. The general structure
of ADRC is shown in Figure 1. In Figure 1 of ADRC, the
transient profile generator is used to obtain each order
derivative 𝑦∗̇ (𝑡), 𝑦∗̈ (𝑡), . . . , 𝑦∗𝑛 (𝑡) of reference trajectory 𝑦∗ (𝑡).
Next, brief description of ADRC is given as follows.
Consider a class SISO nonlinear system as
𝑦𝑛 = 𝑓 (𝑦, 𝑦,̇ . . . , 𝑦(𝑛−1) , 𝑡) + 𝑏𝑢 (𝑡) + 𝑑 (𝑡) .
(1)
Equation (1) also can be described as
𝜀𝑖 = 𝑦∗𝑖 (𝑡) − 𝑧𝑖 ,
..
.
(2)
𝑥𝑛̇ = 𝑓 (𝑥1 , 𝑥2 , . . . , 𝑥𝑛−1 , 𝑡) + 𝑏𝑢 (𝑡) + 𝑑 (𝑡)
𝑦 = 𝑥1 ,
where 𝑓(𝑥1 , 𝑥2 , . . . , 𝑥𝑛−1 , 𝑡) is unknown function, 𝑑(𝑡) is
unknown disturbance, and 𝑢(𝑡) is control input.
Construct the following ESO for nonlinear systems (2):
𝑧̇1 = 𝑧2 − 𝑔1 (𝑧1 − 𝑦)
(3)
𝑧̇𝑛 = 𝑧𝑛+1 − 𝑔𝑛 (𝑧1 − 𝑦) + 𝑏𝑢 (𝑡)
𝑧̇𝑛+1 = − 𝑔𝑛+1 (𝑧1 − 𝑦) .
Let 𝑎(𝑡) = 𝑓(𝑥1 , 𝑥2 , . . . , 𝑥𝑛−1 , 𝑡) + 𝑑(𝑡), so we can obtain
the following conclusion:
𝑧1 󳨀→ 𝑥1 , 𝑧2 󳨀→ 𝑥2 , . . . , 𝑧𝑛 󳨀→ 𝑥𝑛 , 𝑧𝑛+1 󳨀→ 𝑎
(4)
(5)
So the following nonlinear combination can be gotten by
state errors (5):
𝑢0 (𝑡) = 𝑘1 fal (𝜀1 , 𝛼, 𝛿) + ⋅ ⋅ ⋅ + 𝑘𝑛 fal (𝜀𝑛 , 𝛼, 𝛿) ,
(6)
where 𝑘𝑖 , 𝛼, and 𝛿 are adjustable parameters. And nonlinear
function fal is defined as follows:
󵄨󵄨 󵄨󵄨
󵄨󵄨𝜀𝑖 󵄨󵄨 > 𝛿
󵄨󵄨 󵄨󵄨
󵄨󵄨𝜀𝑖 󵄨󵄨 ≤ 𝛿.
(7)
Using the nonlinear state errors feedback (6) and estimation value 𝑎̂(𝑡), the ADRC law can be given by
𝑢 (𝑡) =
..
.
𝑖 = 1, 2, . . . , 𝑛.
󵄨󵄨 󵄨󵄨𝛼
󵄨𝜀 󵄨 sgn (𝜀𝑖 )
{
{󵄨 𝑖 󵄨
fal (𝜀𝑖 , 𝛼, 𝛿) = { 𝜀
{ 𝑖
{ 𝛿1−𝛼
𝑥1̇ = 𝑥2
𝑥𝑛−1 = 𝑥𝑛
through selecting appropriate nonlinear function 𝑔1 , 𝑔2 , . . . ,
𝑔𝑛+1 . Defining that 𝑎̂(𝑡) is the estimation value of 𝑎(𝑡), we can
obtain 𝑧𝑛+1 = 𝑎̂(𝑡).
From the above brief description of ESO, it can be seen
that ESO can be used to estimate the states and the sum of
model uncertainty 𝑓(𝑥1 , 𝑥2 , . . . , 𝑥𝑛−1 , 𝑡) and disturbance 𝑑(𝑡).
So, ESO is such a link, which uses the output 𝑦(𝑡) of plant to
get each order derivative signal 𝑧1 , 𝑧2 , . . . , 𝑧𝑛 and estimation
value of disturbance.
Using 𝑧1 , 𝑧2 , . . . , 𝑧𝑛 from ESO and 𝑦∗̇ (𝑡), 𝑦∗̈ (𝑡), . . . , 𝑦∗𝑛 (𝑡)
from TD, we get the state errors as
𝑢0 (𝑡) − 𝑎̂ (𝑡)
,
𝑏
𝑖 = 1, 2, . . . , 𝑛.
(8)
3. Main Results
Shunt APF circuit schematic is shown in Figure 2; the upper
and lower arm of the shunt APF can be considered as
ideal switch from the APF working principle. The equivalent
circuit of APF is shown in Figure 3. Since the switching
operation can control voltage size of the AC side. So shunt
APF can be considered as a controllable voltage source
and a parallel impedance in the circuit, and to compensate
harmonic current and reactive current can be achieved.
So we can obtain the model of shunt APF as follows:
𝐿
𝑑𝑖𝑐
= 𝑢𝑖 − 𝑅𝑖𝑐 − 𝑢𝑐 .
𝑑𝑡
(9)
Mathematical Problems in Engineering
is
3
iL
where 𝑘11 , 𝑘12 , 𝛼1 , and 𝛿1 are selected parameters. So we can
obtain the ADRC law as
Load
AC
𝑢0 (𝑡) = 𝑘2 fal ((𝑖𝑠∗ (𝑡) − 𝑧1 ) , 𝛼2 , 𝛿2 ) ,
ic
L
𝑢 (𝑡) =
R
APF
Figure 2: Block diagram of shunt APF.
is
iL
Load
AC
ic
S1
L
R
C
S2
+
V
− c
Figure 3: Equivalent circuit of shunt APF.
Define PWM as a proportional part, namely, 𝑢𝑐 = 𝑢𝑉𝑐 ,
where 𝑢 is modulation amount. Let 𝑢 be the control input of
system. 𝑉𝑐 is voltage of DC side. For the supply current, we
know
𝑖𝑠 = 𝑖𝑐 + 𝑖𝐿 .
(10)
Substituting (10) into (9), we have
(𝐿 + 𝐿 𝑠 )
𝑑𝑖𝑠
𝑑𝑖
= − (𝑅 + 𝑅𝑠 ) 𝑖𝑠 − 𝑢𝑉𝑐 + 𝑢𝑠 + 𝑅𝑖𝐿 + 𝐿 𝐿 . (11)
𝑑𝑡
𝑑𝑡
Designed system controller can be considered by a DC
voltage outer-loop control and an inner-loop current control.
Since the response speed of inner-loop is much faster than
the DC voltage outer-loop, it can be considered that DC
voltage is constant when the inner current controls. Ignore
the impedance of the power line; we let 𝑑(𝑡) = 𝑢𝑠 + 𝑅𝑖𝐿 +
𝐿(𝑑𝑖𝐿 /𝑑𝑡); system (11) can be written as
(𝐿 + 𝐿 𝑠 )
𝑑𝑖𝑠
= − (𝑅 + 𝑅𝑠 ) 𝑖𝑠 − 𝑢𝑉𝑐 + 𝑑 (𝑡) .
𝑑𝑡
(12)
The APF is a first-order system. ADRC does not need to
detect the load current and supply voltage and only uses them
as unknown disturbances. A PI controller is used to control
the outer-loop DC voltage, which is order to obtain a given
current value 𝑖𝑠∗ (𝑡). 𝑖𝑠∗ (𝑡) can be seen as the reference input
𝑦∗ (𝑡) of ADRC. The control objective is to make the supply
current 𝑖𝑠 able to track the given current value 𝑖𝑠∗ (𝑡) through
controlling the modulation amount 𝑢 of PWM. Set an order
TD output as
𝑧̇1,1 = −𝑘0 fal ((𝑧1,1 − 𝑖𝑠∗ (𝑡)) , 𝛼0 , 𝛿0 ) ,
(13)
where 𝑘0 , 𝛼0 , and 𝛿0 are selected parameters. Construct of the
following formula ESO:
𝑧̇1 = 𝑧2 − 𝑘11 fal ((𝑧1 − 𝑖𝑠∗ (𝑡)) , 𝛼1 , 𝛿1 ) − 𝑉𝑐 𝑢 (𝑡) ,
𝑧̇2 = −𝑘12 fal ((𝑧1 − 𝑖𝑠∗ (𝑡)) , 𝛼1 , 𝛿1 ) ,
(14)
𝑢0 (𝑡) − 𝑧̇2
,
𝑉𝑐
(15)
where 𝑘2 , 𝛼2 , and 𝛿2 are also selected parameters. All selected
parameters of ADRC controller must try to get in simulation.
RC is mainly used in continuous processes for tracking
or rejecting periodic exogenous signals. In most cases, the
period of the exogenous signal is known. The internal model
principle is the theoretical foundation of RC. According to
internal model principle, to track or reject a certain signal
without steady-state error, the signal can be regarded as the
output of an autonomous generator that is inside the control
system.
Although RC system can still get a good static performance, it cannot get a good dynamic performance of the
system. RC is usually used to meet up with other control
strategies. Actually, RC is only used to restrain the tracking
error. But ADRC can improve the rapid response of the
system. After being coupled with the repetitive controller,
controller can detect the tracking error and accumulate a
correction on the basis of the original command to reduce
the error. Repetitive controller can be seen as an embedded
component, so this system is called embedded repetitive
control system (ERCS). Figure 4 is a block diagram of a
parallel ADRC with RC. Next, how to select the controller
parameters of RC is shown as follows.
(1) Cycle delay factor N: 𝑁 is sampled beat number of
sinusoidal cycle and can be described as fundamental
frequency 𝑓𝑠 and the switching frequency 𝑓𝑐 .
(2) Compensation link 𝑄(𝑧): 𝑄(𝑧) characterizes the
steady precision of repetitive controller. In general,
𝑄(𝑧) is a constant. When 𝑄(𝑧) = 1, the open-loop
gain of system is infinite, and steady-state error is
zero. But this may likely cause system instability. So
we usually select a constant that is less than but close
to 1. 𝑄(𝑧) is also preferably chosen zero phase low
pass filter.
(3) Compensation link 𝑆(𝑧) of plant: 𝑆(𝑧) is used to
reform the controlled plant. After reformation, the
amplitude-frequency characteristics of the plant has
zero gain in the low frequency band. Generally, the
series correction part 𝑆1 (𝑧) is first selected to correct
the low-frequency gain of controlled plant. Then, in
order to improve system stability, the second-order
low-pass filter is selected to attenuate high frequency
gain.
(4) Phase compensation factor 𝑘: the aim of phase compensation factor 𝑘 is to compensate phase lag for
reformed controlled plant in the low frequency.
(5) Repetitive controller gain 𝐾𝑟 : 𝐾𝑟 is used to ensure the
stability of the system in the high frequency band. The
smaller 𝐾𝑟 can cause the better stability, but the speed
of convergence will become slow and the steady-state
4
Mathematical Problems in Engineering
is∗ (t)
Vc∗
−
PI
Kr zk S(z)
z−N
+
Q(z) Z −N
− is (t)
is∗ (t)
k2 fal((is∗ (t) − z1 ), 𝛼2 , 𝛿2 )
u0 (t)
+
1/Vc
u(t)
APF
PWM
−
Vc
is (t)
Vc
z2
ESO of (14)
z1
Figure 4: Block diagram of a parallel ADRC with RC for shunt APF.
50
25
20
10
Electric current (A)
Load electric current (A)
15
5
0
−5
−10
0
−15
−20
−25
0
0.02
0.04
0.06
Time (s)
0.08
0.1
Figure 5: Load electric current.
error will increase. In general, 𝐾𝑟 is chosen to be close
to 1 as possible under maintaining the well stability of
the RC.
4. Simulation Results
In this section, we use Matlab/Simulink for testing and
verifying the proposed APF control method. The parameters
of chosen APF are 𝐿 = 1 H, 𝑅 = 8 Ω, 𝑇 = 100 𝜇s, and
𝑓𝑐 = 10 kHz. The ADRC controller parameters are designed
as 𝛼0 = 2, 𝛼1 = 0.5, 𝛼0 = 1, 𝛿0 = 0.00001, 𝛿1 = 0.001,
𝛿0 = 0.00001, 𝑘0 = 8000, 𝑘11 = 10000, 𝑘12 = 50000, and
𝑘2 = 500. The RC controller parameters are designed as 𝑁 =
200, 𝑆1 (𝑧) = (𝑧 − 0995)/[0.1248(𝑧 − 06)], 𝑆2 (𝑧) = (0.0675𝑧2 +
01349𝑧 + 00675)/(𝑧2 − 1143𝑧 + 0.4128), 𝑄(𝑧) = 0.97, 𝑘 = 3,
and 𝐾𝑟 = 𝑂.8. First, we consider the 150 Hz sine wave for
load. Figures 5 and 6 show the load electric current which is
third harmonic and the output current 𝑖(𝑡) of controlled APF.
−50
0
0.02
0.04
0.06
Time (s)
0.08
0.1
i(t)
i∗ (t)
Figure 6: Reference current 𝑖∗ (𝑡) and output current 𝑖(𝑡) of controlled APF.
Figure 7 shows the total harmonic distortion (THD) analysis
for grid current. It can be seen that the proposed control
method of the APF can better restrain harmonic of grid. The
value of TDH can achieve 0.26%.
First, we consider the 200 Hz square wave signal for load.
Figures 8 and 9 show the load electric current which is
fourth harmonic and the output current 𝑖(𝑡) of controlled
APF. Figure 10 shows the total harmonic distortion (THD)
analysis for grid current. It can be seen that the proposed
control method of the APF can better restrain harmonic of
grid. The value of TDH can achieve 3.88%.
In simulation process, we do not correct the controller
parameters, just change the distortion form of load; the
simulation results are provided to show that the proposed
Mathematical Problems in Engineering
5
FFT analysis
FFT analysis
Fundamental (50 Hz) = 39.54, THD = 3.88%
Fundamental (50 Hz) = 40.09, THD = 0.26%
1.2
Mag (% of fundamental)
Mag (% of fundamental)
4
1
0.8
0.6
0.4
0.2
2
1
0
0
0
5
10
Harmonic order
15
20
0
5
10
Harmonic order
15
20
Figure 10: THD analysis for grid current.
Figure 7: THD analysis for grid current.
Load electric current (A)
3
30
control algorithm of APF has a very reliable robustness and
adaptability for the different distortion forms.
20
5. Conclusions
10
0
−10
−20
−30
0
0.02
0.04
0.06
Time (s)
0.08
0.1
Figure 8: Load electric current.
Since the switch voltage drops, the drive circuit delay differs
and dead zones are affected, there are a large number of
low-order harmonics in the output current of a single-phase
grid-connected APF. The traditional PI controller exists the
capacity deficiencies in the harmonic suppression, and unable
realizes the static error tracking for the sine command
current. It can effectively improve the grid current waveform
through ADRC + RC controller. In this paper, we give the
composite control law design method for single-phase grid
connected APF current loop. Theory and simulation are
provided to show that the proposed control algorithm has
a very reliable tracking ability and satisfactory robustness to
different harmonics of the load.
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Electric current (A)
50
Acknowledgments
The authors wish to thank Professor Yongqiang Ye and Dr. Xu
for their constructive comments and suggestions which have
helped to improve the presentation of the paper. This work
was partially supported by the Advanced Research Project of
Jiangsu Electric Power Company Research Institute.
0
References
−50
0
0.02
0.04
0.06
Time (s)
0.08
0.1
i(t)
i∗ (t)
Figure 9: Reference current 𝑖∗ (𝑡) and output current 𝑖(𝑡) of controlled APF.
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